To generate synthetic data, we also include the effect
of measurement error [1]}. We add Gaussian noise to the time-averaged model statistics, with a diagonal
covariance structure in data space.
We construct the measurement error covariance matrix \(\Delta \)  to be diagonal with entries \(\delta _i>0\) , where \(i\)  indexes over data type (the 3 observed
quantities) and latitude (32 locations). Combining this measurement covariance matrix \(\Delta \)  with the covariance matrix \(\Sigma \)  arising from internal variability leads to an inflated noise covariance matrix
\(\Gamma = \Sigma +\mathop {\mathrm {diag}(\delta _i^2)} = \Sigma + \Delta .\) 
